What is your opinion on the investigation by the FBI about the person who impersonated an FBI officer on Twitter, then? Should people be allowed to call themselves FBI officers?

I have no opinion about US domestic policy and no standing to pronounce one, so I'm unlikely to form one in a hurry just to argue the toss. As for the dumpster fire that is Twitter: the usual advice is, for all Internet denizens is never read the comments, and Twitter is 100% comments. But I digress.

The equivalent, in all my own countries of citizenship, origin, or residence, might be that deceiving someone into thinking you are a police officer is already illegal, although simple impersonation may not be e.g. for satirical purposes. Satire aside, in a country that generally trusts its law enforcement personnel, such a deception, particularly in person, may have both high capacity for harm and a high likelihood it occurs for harm's sake. Nevertheless, notwithstanding my total unfamiliarity with the case you refer to, I struggle to see how Twitter is ever anything other than satire. My own Twitter profile reflects this view.

Curiously, in Australia, it is illegal to claim an association with Sir Donald Bradman, and for similar reasons of potential harm. Impersonating a Prime Minister is not, since it is a very low occupation and any attempt to do so is self-evidently preposterous.